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<div class="subTitle">org.apache.commons.math3.util</div>
<h2 title="Class Precision" class="title">Class Precision</h2>
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<pre>public class <span class="strong">Precision</span>
extends <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre>
<div class="block">Utilities for comparing numbers.</div>
<dl><dt><span class="strong">Since:</span></dt>
  <dd>3.0</dd></dl>
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</ul>
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<caption><span>Fields</span><span class="tabEnd">&nbsp;</span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Field and Description</th>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#EPSILON">EPSILON</a></strong></code>
<div class="block">
 Largest double-precision floating-point number such that
 <code>1 + EPSILON</code> is numerically equal to 1.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#SAFE_MIN">SAFE_MIN</a></strong></code>
<div class="block">Safe minimum, such that <code>1 / SAFE_MIN</code> does not overflow.</div>
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<caption><span>Methods</span><span class="tabEnd">&nbsp;</span></caption>
<tr>
<th class="colFirst" scope="col">Modifier and Type</th>
<th class="colLast" scope="col">Method and Description</th>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static int</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#compareTo(double,%20double,%20double)">compareTo</a></strong>(double&nbsp;x,
         double&nbsp;y,
         double&nbsp;eps)</code>
<div class="block">Compares two numbers given some amount of allowed error.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static int</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#compareTo(double,%20double,%20int)">compareTo</a></strong>(double&nbsp;x,
         double&nbsp;y,
         int&nbsp;maxUlps)</code>
<div class="block">Compares two numbers given some amount of allowed error.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double)">equals</a></strong>(double&nbsp;x,
      double&nbsp;y)</code>
<div class="block">Returns true iff they are equal as defined by
 <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20int)"><code>equals(x, y, 1)</code></a>.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20double)">equals</a></strong>(double&nbsp;x,
      double&nbsp;y,
      double&nbsp;eps)</code>
<div class="block">Returns <code>true</code> if there is no double value strictly between the
 arguments or the difference between them is within the range of allowed
 error (inclusive).</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20int)">equals</a></strong>(double&nbsp;x,
      double&nbsp;y,
      int&nbsp;maxUlps)</code>
<div class="block">Returns true if the arguments are equal or within the range of allowed
 error (inclusive).</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float,%20float)">equals</a></strong>(float&nbsp;x,
      float&nbsp;y)</code>
<div class="block">Returns true iff they are equal as defined by
 <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float,%20float,%20int)"><code>equals(x, y, 1)</code></a>.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float,%20float,%20float)">equals</a></strong>(float&nbsp;x,
      float&nbsp;y,
      float&nbsp;eps)</code>
<div class="block">Returns true if the arguments are equal or within the range of allowed
 error (inclusive).</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float,%20float,%20int)">equals</a></strong>(float&nbsp;x,
      float&nbsp;y,
      int&nbsp;maxUlps)</code>
<div class="block">Returns true if the arguments are equal or within the range of allowed
 error (inclusive).</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(double,%20double)">equalsIncludingNaN</a></strong>(double&nbsp;x,
                  double&nbsp;y)</code>
<div class="block">Returns true if the arguments are both NaN or they are
 equal as defined by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double)"><code>equals(x, y, 1)</code></a>.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(double,%20double,%20double)">equalsIncludingNaN</a></strong>(double&nbsp;x,
                  double&nbsp;y,
                  double&nbsp;eps)</code>
<div class="block">Returns true if the arguments are both NaN, are equal or are within the range
 of allowed error (inclusive).</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(double,%20double,%20int)">equalsIncludingNaN</a></strong>(double&nbsp;x,
                  double&nbsp;y,
                  int&nbsp;maxUlps)</code>
<div class="block">Returns true if both arguments are NaN or if they are equal as defined
 by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20int)"><code>equals(x, y, maxUlps)</code></a>.</div>
</td>
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<tr class="rowColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(float,%20float)">equalsIncludingNaN</a></strong>(float&nbsp;x,
                  float&nbsp;y)</code>
<div class="block">Returns true if both arguments are NaN or they are
 equal as defined by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float,%20float)"><code>equals(x, y, 1)</code></a>.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(float,%20float,%20float)">equalsIncludingNaN</a></strong>(float&nbsp;x,
                  float&nbsp;y,
                  float&nbsp;eps)</code>
<div class="block">Returns true if the arguments are both NaN, are equal, or are within the range
 of allowed error (inclusive).</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsIncludingNaN(float,%20float,%20int)">equalsIncludingNaN</a></strong>(float&nbsp;x,
                  float&nbsp;y,
                  int&nbsp;maxUlps)</code>
<div class="block">Returns true if the arguments are both NaN or if they are equal as defined
 by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float,%20float,%20int)"><code>equals(x, y, maxUlps)</code></a>.</div>
</td>
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<tr class="altColor">
<td class="colFirst"><code>static boolean</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#equalsWithRelativeTolerance(double,%20double,%20double)">equalsWithRelativeTolerance</a></strong>(double&nbsp;x,
                           double&nbsp;y,
                           double&nbsp;eps)</code>
<div class="block">Returns <code>true</code> if there is no double value strictly between the
 arguments or the relative difference between them is less than or equal
 to the given tolerance.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#representableDelta(double,%20double)">representableDelta</a></strong>(double&nbsp;x,
                  double&nbsp;originalDelta)</code>
<div class="block">Computes a number <code>delta</code> close to <code>originalDelta</code> with
 the property that</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#round(double,%20int)">round</a></strong>(double&nbsp;x,
     int&nbsp;scale)</code>
<div class="block">Rounds the given value to the specified number of decimal places.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#round(double,%20int,%20int)">round</a></strong>(double&nbsp;x,
     int&nbsp;scale,
     int&nbsp;roundingMethod)</code>
<div class="block">Rounds the given value to the specified number of decimal places.</div>
</td>
</tr>
<tr class="altColor">
<td class="colFirst"><code>static float</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#round(float,%20int)">round</a></strong>(float&nbsp;x,
     int&nbsp;scale)</code>
<div class="block">Rounds the given value to the specified number of decimal places.</div>
</td>
</tr>
<tr class="rowColor">
<td class="colFirst"><code>static float</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/util/Precision.html#round(float,%20int,%20int)">round</a></strong>(float&nbsp;x,
     int&nbsp;scale,
     int&nbsp;roundingMethod)</code>
<div class="block">Rounds the given value to the specified number of decimal places.</div>
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<h3>Methods inherited from class&nbsp;java.lang.<a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></h3>
<code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#clone()" title="class or interface in java.lang">clone</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#equals(java.lang.Object)" title="class or interface in java.lang">equals</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#finalize()" title="class or interface in java.lang">finalize</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#getClass()" title="class or interface in java.lang">getClass</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#hashCode()" title="class or interface in java.lang">hashCode</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notify()" title="class or interface in java.lang">notify</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#notifyAll()" title="class or interface in java.lang">notifyAll</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#toString()" title="class or interface in java.lang">toString</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait()" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long)" title="class or interface in java.lang">wait</a>, <a href="http://docs.oracle.com/javase/7/docs/api/java/lang/Object.html?is-external=true#wait(long,%20int)" title="class or interface in java.lang">wait</a></code></li>
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<h4>EPSILON</h4>
<pre>public static final&nbsp;double EPSILON</pre>
<div class="block"><p>
 Largest double-precision floating-point number such that
 <code>1 + EPSILON</code> is numerically equal to 1. This value is an upper
 bound on the relative error due to rounding real numbers to double
 precision floating-point numbers.
 </p>
 <p>
 In IEEE 754 arithmetic, this is 2<sup>-53</sup>.
 </p></div>
<dl><dt><span class="strong">See Also:</span></dt><dd><a href="http://en.wikipedia.org/wiki/Machine_epsilon">Machine epsilon</a></dd></dl>
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<pre>public static final&nbsp;double SAFE_MIN</pre>
<div class="block">Safe minimum, such that <code>1 / SAFE_MIN</code> does not overflow.
 <br/>
 In IEEE 754 arithmetic, this is also the smallest normalized
 number 2<sup>-1022</sup>.</div>
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<pre>public static&nbsp;int&nbsp;compareTo(double&nbsp;x,
            double&nbsp;y,
            double&nbsp;eps)</pre>
<div class="block">Compares two numbers given some amount of allowed error.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - the first number</dd><dd><code>y</code> - the second number</dd><dd><code>eps</code> - the amount of error to allow when checking for equality</dd>
<dt><span class="strong">Returns:</span></dt><dd><ul><li>0 if  <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20double)"><code>equals(x, y, eps)</code></a></li>
       <li>&lt; 0 if !<a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20double)"><code>equals(x, y, eps)</code></a> &amp;&amp; x &lt; y</li>
       <li>> 0 if !<a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20double)"><code>equals(x, y, eps)</code></a> &amp;&amp; x > y or
       either argument is NaN</li></ul></dd></dl>
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<pre>public static&nbsp;int&nbsp;compareTo(double&nbsp;x,
            double&nbsp;y,
            int&nbsp;maxUlps)</pre>
<div class="block">Compares two numbers given some amount of allowed error.
 Two float numbers are considered equal if there are <code>(maxUlps - 1)</code>
 (or fewer) floating point numbers between them, i.e. two adjacent floating
 point numbers are considered equal.
 Adapted from <a
 href="http://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/">
 Bruce Dawson</a>. Returns <code>false</code> if either of the arguments is NaN.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>maxUlps</code> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.</dd>
<dt><span class="strong">Returns:</span></dt><dd><ul><li>0 if  <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20int)"><code>equals(x, y, maxUlps)</code></a></li>
       <li>&lt; 0 if !<a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20int)"><code>equals(x, y, maxUlps)</code></a> &amp;&amp; x &lt; y</li>
       <li>&gt; 0 if !<a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20int)"><code>equals(x, y, maxUlps)</code></a> &amp;&amp; x > y
       or either argument is NaN</li></ul></dd></dl>
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<h4>equals</h4>
<pre>public static&nbsp;boolean&nbsp;equals(float&nbsp;x,
             float&nbsp;y)</pre>
<div class="block">Returns true iff they are equal as defined by
 <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float,%20float,%20int)"><code>equals(x, y, 1)</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal.</dd></dl>
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<h4>equalsIncludingNaN</h4>
<pre>public static&nbsp;boolean&nbsp;equalsIncludingNaN(float&nbsp;x,
                         float&nbsp;y)</pre>
<div class="block">Returns true if both arguments are NaN or they are
 equal as defined by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float,%20float)"><code>equals(x, y, 1)</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal or both are NaN.</dd><dt><span class="strong">Since:</span></dt>
  <dd>2.2</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equals(float&nbsp;x,
             float&nbsp;y,
             float&nbsp;eps)</pre>
<div class="block">Returns true if the arguments are equal or within the range of allowed
 error (inclusive).  Returns <code>false</code> if either of the arguments
 is NaN.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>eps</code> - the amount of absolute error to allow.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal or within range of each other.</dd><dt><span class="strong">Since:</span></dt>
  <dd>2.2</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equalsIncludingNaN(float&nbsp;x,
                         float&nbsp;y,
                         float&nbsp;eps)</pre>
<div class="block">Returns true if the arguments are both NaN, are equal, or are within the range
 of allowed error (inclusive).</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>eps</code> - the amount of absolute error to allow.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal or within range of each other,
 or both are NaN.</dd><dt><span class="strong">Since:</span></dt>
  <dd>2.2</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equals(float&nbsp;x,
             float&nbsp;y,
             int&nbsp;maxUlps)</pre>
<div class="block">Returns true if the arguments are equal or within the range of allowed
 error (inclusive).
 Two float numbers are considered equal if there are <code>(maxUlps - 1)</code>
 (or fewer) floating point numbers between them, i.e. two adjacent floating
 point numbers are considered equal.
 Adapted from <a
 href="http://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/">
 Bruce Dawson</a>.  Returns <code>false</code> if either of the arguments is NaN.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>maxUlps</code> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if there are fewer than <code>maxUlps</code> floating
 point values between <code>x</code> and <code>y</code>.</dd><dt><span class="strong">Since:</span></dt>
  <dd>2.2</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equalsIncludingNaN(float&nbsp;x,
                         float&nbsp;y,
                         int&nbsp;maxUlps)</pre>
<div class="block">Returns true if the arguments are both NaN or if they are equal as defined
 by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(float,%20float,%20int)"><code>equals(x, y, maxUlps)</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>maxUlps</code> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if both arguments are NaN or if there are less than
 <code>maxUlps</code> floating point values between <code>x</code> and <code>y</code>.</dd><dt><span class="strong">Since:</span></dt>
  <dd>2.2</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equals(double&nbsp;x,
             double&nbsp;y)</pre>
<div class="block">Returns true iff they are equal as defined by
 <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20int)"><code>equals(x, y, 1)</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal.</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equalsIncludingNaN(double&nbsp;x,
                         double&nbsp;y)</pre>
<div class="block">Returns true if the arguments are both NaN or they are
 equal as defined by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double)"><code>equals(x, y, 1)</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal or both are NaN.</dd><dt><span class="strong">Since:</span></dt>
  <dd>2.2</dd></dl>
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<h4>equals</h4>
<pre>public static&nbsp;boolean&nbsp;equals(double&nbsp;x,
             double&nbsp;y,
             double&nbsp;eps)</pre>
<div class="block">Returns <code>true</code> if there is no double value strictly between the
 arguments or the difference between them is within the range of allowed
 error (inclusive). Returns <code>false</code> if either of the arguments
 is NaN.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - First value.</dd><dd><code>y</code> - Second value.</dd><dd><code>eps</code> - Amount of allowed absolute error.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are two adjacent floating point
 numbers or they are within range of each other.</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equalsWithRelativeTolerance(double&nbsp;x,
                                  double&nbsp;y,
                                  double&nbsp;eps)</pre>
<div class="block">Returns <code>true</code> if there is no double value strictly between the
 arguments or the relative difference between them is less than or equal
 to the given tolerance. Returns <code>false</code> if either of the arguments
 is NaN.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - First value.</dd><dd><code>y</code> - Second value.</dd><dd><code>eps</code> - Amount of allowed relative error.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are two adjacent floating point
 numbers or they are within range of each other.</dd><dt><span class="strong">Since:</span></dt>
  <dd>3.1</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equalsIncludingNaN(double&nbsp;x,
                         double&nbsp;y,
                         double&nbsp;eps)</pre>
<div class="block">Returns true if the arguments are both NaN, are equal or are within the range
 of allowed error (inclusive).</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>eps</code> - the amount of absolute error to allow.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if the values are equal or within range of each other,
 or both are NaN.</dd><dt><span class="strong">Since:</span></dt>
  <dd>2.2</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equals(double&nbsp;x,
             double&nbsp;y,
             int&nbsp;maxUlps)</pre>
<div class="block">Returns true if the arguments are equal or within the range of allowed
 error (inclusive).
 <p>
 Two float numbers are considered equal if there are <code>(maxUlps - 1)</code>
 (or fewer) floating point numbers between them, i.e. two adjacent
 floating point numbers are considered equal.
 </p>
 <p>
 Adapted from <a
 href="http://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/">
 Bruce Dawson</a>. Returns <code>false</code> if either of the arguments is NaN.
 </p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>maxUlps</code> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if there are fewer than <code>maxUlps</code> floating
 point values between <code>x</code> and <code>y</code>.</dd></dl>
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<pre>public static&nbsp;boolean&nbsp;equalsIncludingNaN(double&nbsp;x,
                         double&nbsp;y,
                         int&nbsp;maxUlps)</pre>
<div class="block">Returns true if both arguments are NaN or if they are equal as defined
 by <a href="../../../../../org/apache/commons/math3/util/Precision.html#equals(double,%20double,%20int)"><code>equals(x, y, maxUlps)</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - first value</dd><dd><code>y</code> - second value</dd><dd><code>maxUlps</code> - <code>(maxUlps - 1)</code> is the number of floating point
 values between <code>x</code> and <code>y</code>.</dd>
<dt><span class="strong">Returns:</span></dt><dd><code>true</code> if both arguments are NaN or if there are less than
 <code>maxUlps</code> floating point values between <code>x</code> and <code>y</code>.</dd><dt><span class="strong">Since:</span></dt>
  <dd>2.2</dd></dl>
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<pre>public static&nbsp;double&nbsp;round(double&nbsp;x,
           int&nbsp;scale)</pre>
<div class="block">Rounds the given value to the specified number of decimal places.
 The value is rounded using the <a href="http://docs.oracle.com/javase/7/docs/api/java/math/BigDecimal.html?is-external=true#ROUND_HALF_UP" title="class or interface in java.math"><code>BigDecimal.ROUND_HALF_UP</code></a> method.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Value to round.</dd><dd><code>scale</code> - Number of digits to the right of the decimal point.</dd>
<dt><span class="strong">Returns:</span></dt><dd>the rounded value.</dd><dt><span class="strong">Since:</span></dt>
  <dd>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</dd></dl>
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<pre>public static&nbsp;double&nbsp;round(double&nbsp;x,
           int&nbsp;scale,
           int&nbsp;roundingMethod)</pre>
<div class="block">Rounds the given value to the specified number of decimal places.
 The value is rounded using the given method which is any method defined
 in <a href="http://docs.oracle.com/javase/7/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><code>BigDecimal</code></a>.
 If <code>x</code> is infinite or <code>NaN</code>, then the value of <code>x</code> is
 returned unchanged, regardless of the other parameters.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Value to round.</dd><dd><code>scale</code> - Number of digits to the right of the decimal point.</dd><dd><code>roundingMethod</code> - Rounding method as defined in <a href="http://docs.oracle.com/javase/7/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><code>BigDecimal</code></a>.</dd>
<dt><span class="strong">Returns:</span></dt><dd>the rounded value.</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/ArithmeticException.html?is-external=true" title="class or interface in java.lang">ArithmeticException</a></code> - if <code>roundingMethod == ROUND_UNNECESSARY</code>
 and the specified scaling operation would require rounding.</dd>
<dd><code><a href="http://docs.oracle.com/javase/7/docs/api/java/lang/IllegalArgumentException.html?is-external=true" title="class or interface in java.lang">IllegalArgumentException</a></code> - if <code>roundingMethod</code> does not
 represent a valid rounding mode.</dd><dt><span class="strong">Since:</span></dt>
  <dd>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</dd></dl>
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<pre>public static&nbsp;float&nbsp;round(float&nbsp;x,
          int&nbsp;scale)</pre>
<div class="block">Rounds the given value to the specified number of decimal places.
 The value is rounded using the <a href="http://docs.oracle.com/javase/7/docs/api/java/math/BigDecimal.html?is-external=true#ROUND_HALF_UP" title="class or interface in java.math"><code>BigDecimal.ROUND_HALF_UP</code></a> method.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Value to round.</dd><dd><code>scale</code> - Number of digits to the right of the decimal point.</dd>
<dt><span class="strong">Returns:</span></dt><dd>the rounded value.</dd><dt><span class="strong">Since:</span></dt>
  <dd>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</dd></dl>
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<pre>public static&nbsp;float&nbsp;round(float&nbsp;x,
          int&nbsp;scale,
          int&nbsp;roundingMethod)
                   throws <a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a>,
                          <a href="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</a></pre>
<div class="block">Rounds the given value to the specified number of decimal places.
 The value is rounded using the given method which is any method defined
 in <a href="http://docs.oracle.com/javase/7/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><code>BigDecimal</code></a>.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Value to round.</dd><dd><code>scale</code> - Number of digits to the right of the decimal point.</dd><dd><code>roundingMethod</code> - Rounding method as defined in <a href="http://docs.oracle.com/javase/7/docs/api/java/math/BigDecimal.html?is-external=true" title="class or interface in java.math"><code>BigDecimal</code></a>.</dd>
<dt><span class="strong">Returns:</span></dt><dd>the rounded value.</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathArithmeticException.html" title="class in org.apache.commons.math3.exception">MathArithmeticException</a></code> - if an exact operation is required but result is not exact</dd>
<dd><code><a href="../../../../../org/apache/commons/math3/exception/MathIllegalArgumentException.html" title="class in org.apache.commons.math3.exception">MathIllegalArgumentException</a></code> - if <code>roundingMethod</code> is not a valid rounding method.</dd><dt><span class="strong">Since:</span></dt>
  <dd>1.1 (previously in <code>MathUtils</code>, moved as of version 3.0)</dd></dl>
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<pre>public static&nbsp;double&nbsp;representableDelta(double&nbsp;x,
                        double&nbsp;originalDelta)</pre>
<div class="block">Computes a number <code>delta</code> close to <code>originalDelta</code> with
 the property that <pre><code>
   x + delta - x
 </code></pre>
 is exactly machine-representable.
 This is useful when computing numerical derivatives, in order to reduce
 roundoff errors.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>x</code> - Value.</dd><dd><code>originalDelta</code> - Offset value.</dd>
<dt><span class="strong">Returns:</span></dt><dd>a number <code>delta</code> so that <code>x + delta</code> and <code>x</code>
 differ by a representable floating number.</dd></dl>
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